Three methods of calculating sample size for comparing two counts are available.

The Poisson's Test comparing two counts was initially described by Przyborowski and Wilenski
(see reference), and is known as the Conditional Test (the C Test). The test is based on the
null hypothesis that the ratio of the two count rates (λ2 / λ1) is equal to
1.

Krishnamoorthy and Thomson (see reference) proposed an improvement on the C Test,
where the null hypothesis is that the difference between the two count rates
(λ2 - λ1) is equal to 0. Althought this
test is more complex, the advantages are that it is both more robust and more powerful, so the
sample size required is smaller than that of the C Test.

Whitehead (see reference), in his text book on unpaired sequential analysis, provided
algorithms to determine sample sizes for non-sequential methods, and a method for comparing
two counts was also described.

Sample size calculation for the C Test and the E Test are based on the Poisson distribution.
Computation requires multiple nested loops of calculating the Binomial Coefficient, and the computation time increases
exponentially as the sample size increases. Sample size of more than 100 takes a few seconds
to compute, but sample size over 1000 may take up to 30 minutes to compute.

Whitehead's algorithm is based on a transformation, where the difference between the two
count rates (λ2 - λ1) is assumed to be a mean of a Normally distributed variable.
The calculation is therefore short, and the samle size calculated is smaller than that for both the C Test and the E Test.
This makes the whitehead approach eadier to use, but the assumption of normal distribution becomes
increadingly inappropriate as the sample size decreases.

Please Note : In StatTools, estimating sample size requirement for comparing two counts uses the one tail model.
For a two tail model, do the same calculation using half the α value. For example, sample size for α=0.05 in a
two tail model is the same as that for α=0.1 in the one tail model, everything else being the same.

The plot to the right shows the relationship between sample sizes required calculated from the 3
algorithms.

The x axis represents sample size calculated by Whitehead's algorithm, and the y axis the percentage difference between
sample sizes calculated from the other two algorithms and that from Whitehead's algorithm.

If the sample size calculated by C or E Test is nc/e and sample size calculated by the whitehead algorithm is
nw, the % Difference = (nc/e - nw) / nw x 100.

It can be seen that sample size for the C Test (in blue) is slightly greater than sample size for the E Test (red), as
the E Test is more powerful and so require a smaller sample size.

The sample size of bothe the C and E Tests are greater than that from Whitehead's algorithm, but the difference
(in term of %) decreases as sample size increases.

The differences between the sample sizes from the 3 algorithm are therefore trivial when the sample size is more
than 100, but becomes relevant under that size.

The table of sample size can therefore be consulted, and the sample size required can be derived from numbers in
the table. If the sample size is over 100, it is probably usable. A more precise sample size should be
calculated using the Javascript program, if the initial sample size is estimated to be less than 100.

IntroductionJavascript Program

Please note that computation may take a long time if the sample size is large. On average, sample size less than 100
takes about 30 seconds. Time increases exponentially so that sample size of 200 may take 120 seconds, and further increase may
take even hours.

Please note that some browsers have time limits, and when that limit is reached it asks the user whether to continue or not.
Although long programs can be run, it does require the user to attend and repeatedly tell the browser to continue. The limits are as follows.

This section provides a series of tables presenting commonly used sample sizes comparing two count rates.
The tables are for Type I Error (α) of 0.05 for 1 tail, power of 0.8, and assuming that the two groups have similar sample sizes.

The calculations are from Whitehead, C Test, and E Test, as referenced.

Although the tables present only a limited range of λs, the sample size can be extrapolated from the tables, as,
for the same ratio of the two λs, the sample size is proportionate to the λs, as shown in the following table

λ1

λ2

SSiz by Whitehead

SSiz for C Test

SSiz for E Test

1

2

17

20

19

0.1

0.2

172

197

186

0.01

0.02

1716

1964

1863

0.001

0.002

17158

19637

18631

It can be seen that, as the lambda value decreases by a tenth, the sample size required increases by ten fold.
Sampe size values between cells in the tables can therefore be calculated by extrapolation between the cells.

Sample size for comparing two count rates (λ1 and λ2) with Poisson distribution.